introduction to qcd on the lattice

Why Basics Chall EFTs Rev Future

Introduction to QCD on the lattice

Rainer Sommer

DESY, Zeuthen

September 2005 (DESY, Hamburg)

Why lattice QCD

Basic steps

Challenges

Effective Field Theories

A revolution?

The future

Rainer Sommer Introduction to QCD on the lattice

Why Basics Chall EFTs Rev Future Why Def tool

An Introduction?

I there are about 1.5m of proceedings of annual lattice conferencessince ’85(?)

I there are ≥ 4 text-books

I so what can I say?

if it is not what you need, please ask ...

... or open the Monvay/Munster

An Introduction?

Why do we work on lattice QCD

it provides

I a rigorous definition of QCD

I a computational tool for QCD in the non-perturbative regime−→ numbers for phenomenology

I the connection between perturbative and non-perturbativephenomena

it provides

A rigorous definition of QCD

Wilson regularization

I strict positivity [Luscher;Creutz ]renormalizability (≡ contin. limit) shown to all order of PT [Reisz ]all (flavor) vector symmetries

I axial symmetries have to be restored by proper renormalization [Bochicchio et al. ](as in dim regularization)

Ginsparg Wilson regularization

I exact chiral symmetry in the regularized theory(no questions about γ5)

I can prove– existence of topological suszeptibility [Giusti,Rossi,Testa; Luscher ]– Witten-Veneziano formula– index theorem

I Conceptually clean framework if there ever is a doubt about an n-loopcomputation

A computational tool from “first principles”

I first principles

LQCD = − 1

tr FµνFµν+∑

ψf D + mf ψf

Experiment︷︸︸︷Fπmπ

LQCD(g0,mf )=⇒

QCD parameters (RGI)︷︸︸︷ΛQCD

M = (Mu + Md)/2Ms

Predictions︷︸︸︷ξ

What doesLQCD(g0,mf )

=⇒ mean?

Discretization of LQCD

I gauge invariance

I locality

I unitarity

renormalization ⇓ continuumlimitlow energy matrix elements

√computer time

=⇒ mean?

I gauge invariance

I locality

I unitarity

√computer time

=⇒ mean?

I gauge invariance

I locality

I unitarity

√computer time

=⇒ mean?

I gauge invariance

I locality

I unitarity

√computer time

Why Basics Chall EFTs Rev Future Summ Reg Ren Cont

Basic steps of a computation

this was a bit simplified ... we need the basic steps

I Formulate the problem in the Euclidean

I Discretize ≡ Regularize and simulate

I Renormalize

I Continumm limit (remove the Regularization)

I Renormalize

Formulation

I Easy (for the formulation!) case:

Spectrum and simple matrix elements

an example: flavour currents: −→ time

Aijµ = ψiγµγ5ψj , ipµFK = 〈K (p)|Aus

µ (0)|0〉

∫d3x 〈Aus

0 (x) Asu0 (0)〉 = − 1

2FK2mKe−x0mK

+O(e−(m′K−mK)x0) + O(e−Lmπ )

another example5γ5 γx

s = 2∆

∫d3xd3y 〈Asu

0 (x)O∆s=2(0)Asu0 (y)〉

d3x 〈Aus0 (x)Asu

0 (0)〉∝ BK

Formulation

I Easy (for the formulation!) case:

Spectrum and simple matrix elementsan example: flavour currents:

−→ time

µ (0)|0〉

∫d3x 〈Aus

0 (x) Asu0 (0)〉 = − 1

2FK2mKe−x0mK

s = 2∆

∫d3xd3y 〈Asu

0 (x)O∆s=2(0)Asu0 (y)〉

d3x 〈Aus0 (x)Asu

0 (0)〉∝ BK

Formulation

I Easy (for the formulation!) case:0A A0

Spectrum and simple matrix elementsan example: flavour currents: −→ time

µ (0)|0〉

∫d3x 〈Aus

0 (x) Asu0 (0)〉 = − 1

2FK2mKe−x0mK

s = 2∆

∫d3xd3y 〈Asu

0 (x)O∆s=2(0)Asu0 (y)〉

d3x 〈Aus0 (x)Asu

0 (0)〉∝ BK

Formulation

I Easy (for the formulation!) case:0A A0

Spectrum and simple matrix elementsan example: flavour currents: −→ time

µ (0)|0〉

∫d3x 〈Aus

0 (x) Asu0 (0)〉 = − 1

2FK2mKe−x0mK

s = 2∆

∫d3xd3y 〈Asu

0 (x)O∆s=2(0)Asu0 (y)〉

d3x 〈Aus0 (x)Asu

0 (0)〉∝ BK

Formulation

I easy case: Equilibrium thermodynamicsfinite temperatureEuclidean L0 × L3 spacetime,

ψ(x+L00) = −ψ(x) , ψ(x+L00) = −ψ(x)

Aaµ(x+L00) = +Aa

then path integral = Tre−HL0 = Tre−H/T

L0 = 1/T , T = Temperature

and finite chemical potential µq

LQCD → LQCD + µq ψγ0ψ︸︷︷︸number density

path integral = Tre−(H−µqNq)/T

−→ thermodynamic properties, eg. EOS talk by S. Katz

Formulation

I easy case: Equilibrium thermodynamicsfinite temperatureEuclidean L0 × L3 spacetime,

ψ(x+L00) = −ψ(x) , ψ(x+L00) = −ψ(x)

Aaµ(x+L00) = +Aa

then path integral = Tre−HL0 = Tre−H/T

L0 = 1/T , T = Temperature

and finite chemical potential µq

LQCD → LQCD + µq ψγ0ψ︸︷︷︸number density

path integral = Tre−(H−µqNq)/T

−→ thermodynamic properties, eg. EOS talk by S. Katz

Formulation

I Difficult case: particle decays, scatteringrelation phase shifts ⇔ spectrum in a finite L3 box

[Luscher; Luscher&Wolff; ... ]

I Difficult case: Matrix elements with more than one hadrons ininitial/final statenogo [Maiani & Testa ]go [Lellouche & Luscher ] Matrix elements in a finite L3 box

talks by Giusti, Lubicz

I Impossible (?) casesH H’ → 5 HThe mass of a di-quarkThe gluon condensate...

Formulation

Discretize and “simulate”lattice: xµ = anµ, nµ ∈ ZZ

quarks: ψ(x) on lattice points

gluons: U(x , µ) = P expn

aR 10 dtAµ(x + a(1− t)µ)

o∈ SU(3) on linksEuclidean action: S = SG + SF

tr 1− U(p) ,

SF = a4X

ψ(x)(D(U) + m)ψ(x)

D(U) : discretized Dirac operator

Observables:

〈O〉 = 1Z

ZD[ψ]D[ψ ]D[U] O e−S

Z : 〈1〉 = 1

Can be interpreted as a statistical system in 3 + 1 = 4 dimensions.(S → H/(kT ) , 1/(kT ) ∝ 1/g2

First principles evaluation of 〈O〉 by MC methodstatistical errors ∝ 1/

√computer time.

Renormalization and continuum limit: remove theregularization

For simplicity discussion for Nf = 2 mass-degenerate quarks

I In QCD: critical line with (almost) massless quarks,massless π’s;

Continuum limit: (amhadron → 0) at g0 → 0

Along RGT ≡ fixed physical quark mass:

a mhadron ∼ Chadron e−1/(2b0g20 )(2b0g

20 )−b1/2b2

0 × [1 + O(g20 )]| z

badly convergent

b0 = (11− 23Nf)/(4π)2

I More precisely

fix m2π/m

2nucleon = 137/938← experiment

fix mnucleon = 938 MeV← experiment

⇒ a =a mnucleon

938MeV← “setting the scale”

other hadron masses and e.g. Fπ are predictions up to O(a) or O(a2) errors

bare parametersg0,m0

→ renormalized onesmπ ,mnucleon

← hadronicrenormalization scheme

I Equivalently:

fix m2π/m

2nucleon, then

mhadron

mnucleon

˛lattice

=mhadron

mnucleon

× [1 + O(an)]

I often: “know” that one is close to the continuum limit

I More precisely

fix m2π/m

⇒ a =a mnucleon

I Equivalently:

fix m2π/m

2nucleon, then

mhadron

mnucleon

˛lattice

=mhadron

mnucleon

× [1 + O(an)]

I More precisely

fix m2π/m

⇒ a =a mnucleon

I Equivalently:

fix m2π/m

2nucleon, then

mhadron

mnucleon

˛lattice

=mhadron

mnucleon

× [1 + O(an)]

I More precisely

fix m2π/m

⇒ a =a mnucleon

I Equivalently:

fix m2π/m

2nucleon, then

mhadron

mnucleon

˛lattice

=mhadron

mnucleon

× [1 + O(an)]

I better: see

expl. from [Garden et al. 1999 ] quenched

I better: see

expl. from [Garden et al. 1999 ] quenched

Continuum limit continued

I bewareof phase transitions

example fromtmQCD[Sharpe et al.; Munster;

Scorzato ][Farchioni et al., χLF Coll. ]

κ = (2µκ)-1

β1 1/g02

cont.limit

Why Basics Chall EFTs Rev Future Summ hmc c&b c&b Ren pow corr

The challenges for lattice QCD as a computational tool

large L, small a, mquark of different scales

I numerical cost of a computation

I renormalization and mixing

I heavy quarks → Lubicz

furthermore:

I short physical distances to connect to PT → Knechtli

I complex actions (finite µq) very difficult for MC-technique → Katz(Seff(U) is not real for finite µq)

furthermore:

The work horse of LQCD: the HMC [Duane, Kennedy, Pendleton, Roweth ]

〈O(Φ)〉 = 1Z

Oe−S(Φ) Φ = U(x , µ), ψ(x), ψ(x)

O(Φi ) for P(Φ) ∝ e−S(Φ) > 0

local bosonic actions: Φi with Metropolis, heatbath, OR algo’s: fast

QCD: integrate out fermions (Grassmann)

〈O(U)〉 = 1Z

[O(U)]F exp(−SG(U) + Nf Tr ln(D(U) + m0)| z non−local

rewrite with bosonic field φ

〈O(U)〉 = 1Z

ZU,φ,φ†

[O(U)]F exp(−SG(U)− Nf2φ†[(D(U) + m0)

†(D(U) + m0)]−1φ)

≡ 1Z

ZU,φ,φ†

[O(U)]Fe−SPF

〈O(Φ)〉 = 1Z

Oe−S(Φ) Φ = U(x , µ), ψ(x), ψ(x)

O(Φi ) for P(Φ) ∝ e−S(Φ) > 0

〈O(U)〉 = 1Z

ZU,φ,φ†

†(D(U) + m0)]−1φ)

≡ 1Z

ZU,φ,φ†

[O(U)]Fe−SPF

〈O(Φ)〉 = 1Z

Oe−S(Φ) Φ = U(x , µ), ψ(x), ψ(x)

O(Φi ) for P(Φ) ∝ e−S(Φ) > 0

〈O(U)〉 = 1Z

ZU,φ,φ†

†(D(U) + m0)]−1φ)

≡ 1Z

ZU,φ,φ†

[O(U)]Fe−SPF

introduce a (classical) hamiltonian with U as coordinates and SPF as potential andconjugate variables to U (momenta p)

H =Xx,µ

12p2(x , µ) + SPF(U, φ)

algorithm

I p, φ with P(φ) ∝ e−SPF(U,φ), P(p) ∝ e−12

I Hamiltonian evolution

(of course discretisedwith step-size δt)

dU= −−dSPF

I U′; metropolis accept/reject (corrects step size errors)

−→ P(U) ∝ e−Seff (U)

H =Xx,µ

12p2(x , µ) + SPF(U, φ)

algorithm

dU= −−dSPF

H =Xx,µ

12p2(x , µ) + SPF(U, φ)

algorithm

dU= −−dSPF

I the expensive step

−dSPF

dU= Nf

2φ†[(D(U) + m0)

†(D(U) + m0)]−1 dD

dU(D(U) + m0)

−1φ)

1. inversion of operator is expensive ∼ V4m−1q

2. dSPFdU→ large for small mq

−→ small δt in discretization of evolution

effort ∝ (δt)−1 ∝ m−nq , n>∼2

total effort ∝ V4 m−nq , n>∼3

(in finite volume, L4, (with Dirichlet B.C.’s) total effort ∝ (L/a)−(4+n) , n>∼3

independent of mq SF simulations −→ Knechtli)

I in general: small quark masses are very difficult to reach

I reach up+down quark masses from heavier onesguided by ChPT → later

−dSPF

dU= Nf

2φ†[(D(U) + m0)

†(D(U) + m0)]−1 dD

dU(D(U) + m0)

−1φ)

−dSPF

dU= Nf

2φ†[(D(U) + m0)

†(D(U) + m0)]−1 dD

dU(D(U) + m0)

−1φ)

−dSPF

dU= Nf

2φ†[(D(U) + m0)

†(D(U) + m0)]−1 dD

dU(D(U) + m0)

−1φ)

−dSPF

dU= Nf

2φ†[(D(U) + m0)

†(D(U) + m0)]−1 dD

dU(D(U) + m0)

−1φ)

−dSPF

dU= Nf

2φ†[(D(U) + m0)

†(D(U) + m0)]−1 dD

dU(D(U) + m0)

−1φ)

The costs: e.g. for a = 0.08 fm, 1000 configurations Ui

(0.05 fm a factor ∼30)

I Berlin “wall” ——–Lattice 2001, Berlin

disappeared through

Mass preconditioning(“Hasenbusch trick”)[Hasenbusch; Jansen & Hasenbusch; ... ]

multiple time scales[Urbach et al., 2005 ]

Tflops · years

staggered ref [13]

ref [12]

this work

mPS/mV

I more radical: domain decomposition [Luscher ] −→ Giustiprofiting from locality of QCD

disappeared through

Tflops · years

staggered ref [13]

ref [12]

this work

mPS/mV

disappeared through

Tflops · years

staggered ref [13]

ref [12]

this work

mPS/mV

The quenched approximation

〈O(U)〉 = 1Z

[O(U)]F| z [D(U)+m0]−1

exp(−SG(U) + Nf Tr ln(D(U) + m0)| z non−local drop it

Nf → 0 limit, no sea quarks−→

I quenchedhadron mass spectrumcont. limt[CP-PACS,2002 ]

mh [GeV]

ΞΣ ∆

input: mπ, mρ, mφ

I Amazing success, but qQCD is not QCD. typically 90+% of the answerFactor > 100 work for the remaining 10-x% no better way?

〈O(U)〉 = 1Z

[O(U)]F| z [D(U)+m0]−1

mh [GeV]

ΞΣ ∆

〈O(U)〉 = 1Z

[O(U)]F| z [D(U)+m0]−1

mh [GeV]

ΞΣ ∆

Heavy quarks: The particular challenge for B-physics

multiple scale problemalways difficultfor a numerical treatment

I Take a large lattice as it is possible in the quenched approximation

λπ = 1/mπ ≈ L

λB = 1/mb < a

light quarks are too light→ treat by an extrapolation

b-quark is too heavy

I Need an effective theory for the b-quark: HQET, NRQCD[E. Eichten, 1988; E. Eichten & B. Hill 1990; Caswell & Lepage; Lepage & Thacker ]

λπ = 1/mπ ≈ L

λB = 1/mb < a

λπ = 1/mπ ≈ L

λB = 1/mb < a

I b-quark istoo heavy:

aMb ≈ 4aMc !

charm: [Sint & Rolf ]

3 definitions ofcharm quark massdiffer by a-effects

charm just doable

a possible alternative(under investiga-tion): Symanzikimprovement toall orders in am[Aoki,Kuramashi ]∼ Fermilab action [El

Khadra,Kronfeld,Mackenzie

I for b-quarks a→ 0 can’t be controlled in this way

I b-quark istoo heavy:

aMb ≈ 4aMc !

charm: [Sint & Rolf ]

3 definitions ofcharm quark massdiffer by a-effects

charm just doable

a possible alternative(under investiga-tion): Symanzikimprovement toall orders in am[Aoki,Kuramashi ]∼ Fermilab action [El

Khadra,Kronfeld,Mackenzie

I for b-quarks a→ 0 can’t be controlled in this way

Renormalization (with mixing)

1. Bare parameters of QCD, g0,m0,i

→ fundamental parameters, Λ,M0,i ( RGI’s).

prediction of Λ,M0,i with hadron spectrum as input. −→ Knechtli

2. Renormalization of composite operators

e.g. 〈K0|Hweak|K0〉SM ↔ C(µ)〈K0|O∆s=2(µ)|K0〉QCD

↑matching

similar

〈B|Abu0 |0〉 ↔ C(µ)〈B|Abu,stat

0 (µ)|0〉+ O(1/mb) , µ = mb

I matching usually done in the continuum, dimensional regularization,MS-renormal. of coupling and massesstill have to renormalize the bare lattice operators and connect to MS

3. Weak currents(AR)µ = ZA(g0)Aµ

I special case of composite operators: renormalization is scaleindependent

I can be fixed by χ WI’s [Bochicchio et al. ]I ZA = 1 if the action has exact (lattice) χ symmetry

4. Coefficients in effective (lattice) actions

I e.g. lattice HQET5. Others, in particular hard problems like

OR = Z(µ)

c1(g0)

c2(g0)

a2O2 +

c3(g0)

ffI e.g. ∆I = 1/2 problem

1.A Take a physical observable, expandable in renormalized perturbation theory, e.g.

F (r) =4

˘αMS(µ) + c1[αMS(µ)]2 + . . .

¯, µ = 1/r

r2αqq(µ) physical coupling

compute the physical coupling for large µ; → Λin practice: the Schrodinger functional → Knechtli

1.B Use bare (UV) quantity, expandable in bare (lattice) perturbation theory

e.g. P = 1N〈 trU(p)〉 α ≡ − 1

CFπln( P|z

from MC!

then αMS(s0a−1) = α + 0.614α3

+ O(α40) + O(a)

or (see later) αV

properties

A) “physical” observable

+ only continuum PT

+ separates continuum limit (an) and renormalization effects (αm)

– needs “window”L−1 Λ µ a−1

+ modification L−1 ≡ µ (use the finite volume); then L a issufficient;large µ by recursion µ0 → 2µ0 → . . .→ 2nµ0

B) UV (cutoff) quantity

+ easy, quick (all you need is a hadron mass)

– mixing of discretization errors (an) and renormalization effects (αm):αMS(s0a

– difficult (for Nf > 0 impossible) to reach high µ

2. Renormalization of composite operatorsExample

O(x) ≡ O∆s=2(x) = s(x)γLµd(x) s(x)γL

µd(x)

assume exact χ symmetry → OR(x) = ZO (µ)O(x)a possible renormalization condition, intermediate scheme

(P(x) = d(x)γ5s(x), P+(x) = s(x)γ5d(x))

〈PR(x1)OR(0)PR(x2)〉 = Z2P(µ)ZO (µ)〈P(x1)O(0)P(x2)〉 e.g. x1 = −x2

= 〈P(x1)O(0)P(x2)〉tree level x21 = x2

2 = µ−2 = x2

〈PR(x)P+R(0)〉 = Z2

P(µ)〈P(x)P(0)〉 = 〈P(x)P(0)〉tree level

? gauge invariant

? for large µ connected to other schemes

OR(µ) = OMS(µ)× [1 + c1αMS + . . .]

? problems of such a ren. cond.– signals in MC– need PT (c1 . . .)– need window: L−1 Λ µ a−1

→ not practical

alternative 1:

? fix gauge

? P(x) = d(x)γ5s(x)→ d(y1)s(y2) “quark states”

? go to momentum space → MOM-scheme (“RI-MOM”)

– on-shell improvement does not work

– off-shell improvement with non-gaugeinvariant terms

+ rather simple, “universal”

alternative 2:

? renormalization in finite volume, in particular in the SFL a is sufficient;large µ by recursion

+ in the SF: use boundary quark fields instead of P(x)

+ gauge invariant

+ on-shell improvement does work

– need to do PT ourselves (usually 1-loop, sometimes 2)

alternative 1:

? fix gauge

? P(x) = d(x)γ5s(x)→ d(y1)s(y2) “quark states”

? go to momentum space → MOM-scheme (“RI-MOM”)

– on-shell improvement does not work

– off-shell improvement with non-gaugeinvariant terms

+ rather simple, “universal”

alternative 2:

? renormalization in finite volume, in particular in the SFL a is sufficient;large µ by recursion

+ in the SF: use boundary quark fields instead of P(x)

+ gauge invariant

+ on-shell improvement does work

– need to do PT ourselves (usually 1-loop, sometimes 2)

4. Coefficients in effective (lattice) actions

Coefficients in effective (lattice) actions e.g. lattice HQET

SQCD(. . . ,M) ↔ SHQET(. . . , cσ·B(g0), c2D(g0), . . .)

↑matching

Matching, in principle (M = mb):

OQCD(. . . ,M) = OHQET(. . . , cσ·B(g0), c2D(g0), . . .)

+O(( |p|M )n, ( ΛM )n) + O(am)

This may be done non-perturbatively [Heitger, S. ](again using a finite volume for the matching step)

Mixing of operators of different dimension

appears e.g. in effective theories such as HQET

OR = Z (µ)

perturbation theory is not enough for c , d :

PT: c = c1 g20 + c2g

40 + ...+ cng

2n0 (take d = 0)

I perturbative uncertainty in OR:

δOR ∼g2n+20

a∼ 1

2b0 ln(1/aΛ)n+1 a→∞

no continuum limit

I it is a more general QCD problem: clean computation ofpower corrections needs full non-perturbative treatment of theleading term

Mixing of operators of different dimension

appears e.g. in effective theories such as HQET

OR = Z (µ)

perturbation theory is not enough for c , d :

PT: c = c1 g20 + c2g

40 + ...+ cng

2n0 (take d = 0)

I perturbative uncertainty in OR:

δOR ∼g2n+20

a∼ 1

2b0 ln(1/aΛ)n+1 a→∞

no continuum limit

I it is a more general QCD problem: clean computation ofpower corrections needs full non-perturbative treatment of theleading term

And it is possible: e.g. 1/m corrections in HQET

it is possible with full non-perturbative renormalization

experiment Lattice with amq 1

mB = 5.4 GeV Φ1(L1,M),Φ2(L1,M)

ΦHQET1 (L2),Φ

HQET2 (L2) ΦHQET

1 (L1),ΦHQET2 (L1)σm(u1)

σkin1 (u1), σ

kin2 (u1)

L2 = 2L1

[ LPHAACollaboration : Heitger & S; Heitger & Wennekers; Heitger, Juttner,S & Wennekers ]

very recent (quenched!): mb, MS-scheme, cont.lim.

mstatb (mb) = 4350(64) MeV ← ±O(Λ2/mb)

mstatb (mb) + m1

b(mb) = 4301(70) MeV ← ±O(Λ3/m2b)

[ LPHAACollaboration : Della Morte, Garron, Papinutto, S. ]

And it is possible: e.g. 1/m corrections in HQET

it is possible with full non-perturbative renormalization

experiment Lattice with amq 1

mB = 5.4 GeV Φ1(L1,M),Φ2(L1,M)

ΦHQET1 (L2),Φ

HQET2 (L2) ΦHQET

1 (L1),ΦHQET2 (L1)σm(u1)

σkin1 (u1), σ

kin2 (u1)

L2 = 2L1

[ LPHAACollaboration : Heitger & S; Heitger & Wennekers; Heitger, Juttner,S & Wennekers ]

very recent (quenched!): mb, MS-scheme, cont.lim.

mstatb (mb) = 4350(64) MeV ← ±O(Λ2/mb)

mstatb (mb) + m1

b(mb) = 4301(70) MeV ← ±O(Λ3/m2b)

[ LPHAACollaboration : Della Morte, Garron, Papinutto, S. ]

Why Basics Chall EFTs Rev Future Summ Sym WChPT

Effective field theories for (lattice) QCD

|pµ| Mlow energy, small momentum expansionsderivative expansions e.g.LHQET = ψhD0ψh + 1

mbψhDkDkψh

I M = mb HQET

I M = a−1 Symanzik effective theory

I M = T = Temperature dimensional reduction

I M = 4πFπ and B ×mquark = O(|pµ|2), B = −〈ψψ〉/F 2π ChPT

renormalizable order by order in the expansion → will yield( finite number of counterterms) the asymptotic expansion in |pµ|/Mvery useful (even necessary?)

new developments: combination of ChPT and Symanzik effective theory:

WChPT, SChPT

Symanzik effective theory

At energies, momenta far below the cutoff 1/a, lattice QCD is equivalent to acontinuum theory with effective action (on shell)

Seff = SQCD + aS1 + a2S2 + . . .

Zd4x c(g0) ψ σµνFµνψ + less relevant

Zd4x c ′(g0) trDρFµνDρFµν + ...

⇒ O(a) cutoff effects in on-shell matrix elements can be canceled by adding

i4cswψσµνFµνψ, csw = 1 + O(g2

to the lattice action [Symanzik, Luscher & Weisz, Sheikholeslami & Wohlert,..., LPHAACollaboration ]

I ψ σµνFµνψ is not chirally symmetric (so is the Wilson action)

I other actions exist, which have enough chiral symmetry to enforce c(g0) = 0“automatic O(a) improvement”

Seff = SQCD + aS1 + a2S2 + . . .

The Zoo of fermion actionsI Wilson: SUL(Nf)× SUR(Nf)× UV (1) broken → SUV (Nf)× UV (1)

O(a)-effectsχ symmetry only in the continuum limit keep m0 > O(aΛ2)mixing of operators of different dimension and of different chirality

I O(a)-improved Wilson: the same but O(a2) effects −→[Sheikholeslami & Wohlert; ...; LPHAA

Collaboration ]I tmQCD: mass term (per doublet) mψγ5τ

3ψ [Frezzotti, Grassi, Sint & Weisz ]simplified O(a) improvementsolves many mixing problemsbreaks parity to PF breaks flavor symmetryat maximum twist: automatic O(a) improvement: [Frezzotti & G.C.Rossi ]

I Domain Wall fermions [Kaplan; Furman & Shamir ]good χ symmetry expensive (factor ≈ 20)

I classically perfect [Hasenfratz & Niedermayer ]good χ symmetry expensive (factor ≈ 10− 50)small a-effects −→

I Ginsparg-Wilson type [Laliena, Hasenfratz, Niedermayer; Neuberger ]exact χ symmetry [Luscher ] very expensive (factor ≈ 10− 100)automatic O(a) improvement

I staggered [Kogut&Susskind ] 4 tastes, but often reduced to 1 by “4-th root trick”(vector) flavor symmetry broken (pions with different masses)

I highly improved; FLIC, ...

Chiral perturbation theory including a-effectslow energy effective Lagrangian for QCD: χ Lagrangian

[Weinberg; Gasser & Leutwyler ]

L = L2 + L4 + . . .

L2 =F 2

“∂µΣ†∂µΣ

“χ†Σ + χΣ†

”L4 = . . .+ L4 Tr

“∂µΣ†∂µΣ

“Σ†χ+ χ†Σ

”+L5 Tr

h∂µΣ†∂µΣ

“Σ†χ+ χ†Σ

”i+L6

“Σ†χ+ χ†Σ

”i2+ L7

“Σ†χ− χ†Σ

+L8 Tr“χ†Σχ†Σ + χΣ†χΣ†

”+ . . .+ L12 Trχχ†

M ≡ diag(mu ,md ,ms) Σ ≡ exp

„i2Taπa

«χ ≡ 2MB

and a− terms: aψ σµνFµνψ → Tr“∂µΣ†∂µΣ

“Σ†ρ+ ρ†Σ

”+ . . .

ρ = a cρ(g0) 1

[Sharpe & Singleton 1998; Sharpe & Shoresh 2002; Rupak & Shoresh 2002 ]

and a− terms: aψ σµνFµνψ → tr(∂µΣ†∂µΣ

(Σ†ρ+ ρ†Σ

)+ . . .

ρ = a cρ(g0) 1

Predicts, for mq ΛQCD , a 1/ΛQCD , e.g. (quarks degenerate)

µ= 1 + γ︸︷︷︸

logµ+ α︸︷︷︸Li

+ α′︸︷︷︸Li

ρ+ ω︸︷︷︸Wi

ρ+ O(µ2, µρ, ρ2)

prediction low energy constants: from fits

µ = Bmq, ρ = acρ(g0)

when mq, a are small enough:

functional formfor extrapolations

mq → mu,md ,

a → 0is known

There are of course extensions to O(a2) very active right now.

A recent revolution?

I some transparencies taken from C. DaviesHEP2005 International Europhysics Conference on High Energy Physics

Lattice QCD 2005Christine Davies

University of Glasgow

Lattice 2005 happening in Dublin

See www.tcd.ie/lat05/index.php

HEPP-EPS 2005

A recent revolution?I

Take-home message

¥ There has been a revolution in lattice QCD

since 2003

¥ Quenched approximation (ignores sea

quarks) is dead - stop quoting results from it

¥ Lattice QCD now delivering fully

unquenched results: hadron masses that

agree with expt; precise parameters of QCD;

matrix elements relevant to CKM physics.

IImproved staggered quarks do not solve doubling problem

2 4 6 8 10 12 14 16

Wilson Glue (a ≈

o e Glue (a ≈

FAT7xASQ

2 4 6 8 10 12 14 16

Wilson Glue (a ≈

o e Glue (a ≈

FAT7xASQ

ASQTAD

2 4 6 8 10 12 14 16

Wilson Glue (a ≈

o e Glue (a ≈

FAT7xASQ

ASQTAD

ONE-LINK

2 4 6 8 10 12 14 16

Wilson Glue (a ≈

o e Glue (a ≈

FAT7xASQ

ASQTAD

ONE-LINK

2 4 6 8 10 12 14 16

EIGENVALUES

i en alues

Wilson Glue (a ≈

o e Glue (a ≈

2 4 6 8 10 12 14 16

Wilson Glue (a ≈

o e Glue (a ≈

2 4 6 8 10 12 14 16

Wilson Glue (a ≈

o e Glue (a ≈

2 4 6 8 10 12 14 16

CHIRALITY

Wilson Glue (a ≈

o e Glue (a ≈

FAT7xASQ

2 4 6 8 10 12 14 16

CHIRALITY

Wilson Glue (a ≈

o e Glue (a ≈

FAT7xASQ

ASQTAD

2 4 6 8 10 12 14 16

CHIRALITY

Wilson Glue (a ≈

o e Glue (a ≈

FAT7xASQ

ASQTAD

ONE-LINK

2 4 6 8 10 12 14 16

CHIRALITY

Wilson Glue (a ≈

o e Glue (a ≈

FAT7xASQ

ASQTAD

ONE-LINK

2 4 6 8 10 12 14 16

EIGENVALUES

Wilson Glue (a ≈

o e Glue (a ≈

2 4 6 8 10 12 14 16

EIGENVALUES

Wilson Glue (a ≈

o e Glue (a ≈

2 4 6 8 10 12 14 16

EIGENVALUES

Wilson Glue (a ≈

o e Glue (a ≈

2 4 6 8 10 12 14 16

CHIRALITY

Wilson Glue (a ≈

o e Glue (a ≈

FAT7xASQ

2 4 6 8 10 12 14 16

CHIRALITY

Wilson Glue (a ≈

o e Glue (a ≈

FAT7xASQ

ASQTAD

2 4 6 8 10 12 14 16

CHIRALITY

Wilson Glue (a ≈

o e Glue (a ≈

FAT7xASQ

ASQTAD

ONE-LINK

2 4 6 8 10 12 14 16

CHIRALITY

Wilson Glue (a ≈

o e Glue (a ≈

FAT7xASQ

ASQTAD

ONE-LINK

2 4 6 8 10 12 14 16

EIGENVALUES

Wilson Glue (a ≈

o e Glue (a ≈

2 4 6 8 10 12 14 16

EIGENVALUES

Wilson Glue (a ≈

o e Glue (a ≈

2 4 6 8 10 12 14 16

EIGENVALUES

Wilson Glue (a ≈

o e Glue (a ≈

CHIRALITY

Follana et al (2004-5). . G.P. Lepage, High-Precision LQCD (April 2005). Ð p. 21/43

4 ÔtastesÕ of quark instead of 1 - if all same, Ôdivide by 4Õ :

det(M) → det1/4(M)

ÔTaste-changingÕ interactions mess this up, but vanish as a2

NEW theoretical work encouraging, e.g. eigenvalues

divides into quartets, Index Theorem obeyed etc.

Follana et al; Durr et al; Adams; Peardon et al; Shamir

I No Lagrangian formulation known for this “fourth root trick”a discussion in the lattice community:

Q: what are the quark fields?Q: is there a continuum limt? is it QCD?

A: comparison to experiment Q: a good model?

IImproved staggered quarks do not solve doubling problem

2 4 6 8 10 12 14 16

Wilson Glue (a ≈

o e Glue (a ≈

FAT7xASQ

2 4 6 8 10 12 14 16

Wilson Glue (a ≈

o e Glue (a ≈

FAT7xASQ

ASQTAD

2 4 6 8 10 12 14 16

Wilson Glue (a ≈

o e Glue (a ≈

FAT7xASQ

ASQTAD

ONE-LINK

2 4 6 8 10 12 14 16

Wilson Glue (a ≈

o e Glue (a ≈

FAT7xASQ

ASQTAD

ONE-LINK

2 4 6 8 10 12 14 16

EIGENVALUES

i en alues

Wilson Glue (a ≈

o e Glue (a ≈

2 4 6 8 10 12 14 16

Wilson Glue (a ≈

o e Glue (a ≈

2 4 6 8 10 12 14 16

Wilson Glue (a ≈

o e Glue (a ≈

2 4 6 8 10 12 14 16

CHIRALITY

Wilson Glue (a ≈

o e Glue (a ≈

FAT7xASQ

2 4 6 8 10 12 14 16

CHIRALITY

Wilson Glue (a ≈

o e Glue (a ≈

FAT7xASQ

ASQTAD

2 4 6 8 10 12 14 16

CHIRALITY

Wilson Glue (a ≈

o e Glue (a ≈

FAT7xASQ

ASQTAD

ONE-LINK

2 4 6 8 10 12 14 16

CHIRALITY

Wilson Glue (a ≈

o e Glue (a ≈

FAT7xASQ

ASQTAD

ONE-LINK

2 4 6 8 10 12 14 16

EIGENVALUES

Wilson Glue (a ≈

o e Glue (a ≈

2 4 6 8 10 12 14 16

EIGENVALUES

Wilson Glue (a ≈

o e Glue (a ≈

2 4 6 8 10 12 14 16

EIGENVALUES

Wilson Glue (a ≈

o e Glue (a ≈

2 4 6 8 10 12 14 16

CHIRALITY

Wilson Glue (a ≈

o e Glue (a ≈

FAT7xASQ

2 4 6 8 10 12 14 16

CHIRALITY

Wilson Glue (a ≈

o e Glue (a ≈

FAT7xASQ

ASQTAD

2 4 6 8 10 12 14 16

CHIRALITY

Wilson Glue (a ≈

o e Glue (a ≈

FAT7xASQ

ASQTAD

ONE-LINK

2 4 6 8 10 12 14 16

CHIRALITY

Wilson Glue (a ≈

o e Glue (a ≈

FAT7xASQ

ASQTAD

ONE-LINK

2 4 6 8 10 12 14 16

EIGENVALUES

Wilson Glue (a ≈

o e Glue (a ≈

2 4 6 8 10 12 14 16

EIGENVALUES

Wilson Glue (a ≈

o e Glue (a ≈

2 4 6 8 10 12 14 16

EIGENVALUES

Wilson Glue (a ≈

o e Glue (a ≈

CHIRALITY

Follana et al (2004-5). . G.P. Lepage, High-Precision LQCD (April 2005). Ð p. 21/43

4 ÔtastesÕ of quark instead of 1 - if all same, Ôdivide by 4Õ :

det(M) → det1/4(M)

ÔTaste-changingÕ interactions mess this up, but vanish as a2

NEW theoretical work encouraging, e.g. eigenvalues

divides into quartets, Index Theorem obeyed etc.

Follana et al; Durr et al; Adams; Peardon et al; Shamir

I No Lagrangian formulation known for this “fourth root trick”a discussion in the lattice community:

Q: what are the quark fields?Q: is there a continuum limt? is it QCD?

A: comparison to experiment Q: a good model?

IUpdate on light decay constants

Improved chiral

extraps for

f!= 1.200(4)(+17

−5 )

Vus = 0.2238(+12−32)

CKM 2005 world

average: 0.2262(23)(Blucher)

Use staggered chiral pert. th. to extrap. to physical pion mass (Aubin + Bernard) to take account of disc. errors. NEW 2005 - configswith second ms.

f!, fK

MILC, Aubin et al, hep-lat/0407028; Heller,

this meeting, Bernard LAT05.

!(K → µ") =G2FmKm

8#(1−

m2K) f 2KV

Assuming Vud

I fit with 20 free parametersm2

q terms and many terms from StaggeredChPT

IUpdate on light decay constants

Improved chiral

extraps for

f!= 1.200(4)(+17

−5 )

Vus = 0.2238(+12−32)

CKM 2005 world

average: 0.2262(23)(Blucher)

Use staggered chiral pert. th. to extrap. to physical pion mass (Aubin + Bernard) to take account of disc. errors. NEW 2005 - configswith second ms.

f!, fK

MILC, Aubin et al, hep-lat/0407028; Heller,

this meeting, Bernard LAT05.

!(K → µ") =G2FmKm

8#(1−

m2K) f 2KV

Assuming Vud

I fit with 20 free parametersm2

q terms and many terms from StaggeredChPT

Concerning StaggeredChPTI Necessary because one staggered field describes 4 quarks, but flavor symmetry is

broken

FIG. 9: Squared masses of charged pions for various tastes on the coarse lattices. We use r1 to set

the scale. Tastes that are degenerate by SO(4) symmetry are fit together.

r1 ∼ 1/600MeV−→ splittings of450 MeV

SChPT applicable?with the quoted preci-sion?

Concerning StaggeredChPTI Necessary because one staggered field describes 4 quarks, but flavor symmetry is

broken

FIG. 9: Squared masses of charged pions for various tastes on the coarse lattices. We use r1 to set

the scale. Tastes that are degenerate by SO(4) symmetry are fit together.

r1 ∼ 1/600MeV−→ splittings of450 MeV

SChPT applicable?with the quoted preci-sion?

IDetermining Parameters of QCD :

nf = 3

nf = 00.2

V(d/a)

2 4 6 8

d/a(G eV)

Combine 3-loop

pert. th. for 28

different

quantities (mainly

Wilson loops)

with numerical

calc. on lattice

Results at 3 values of a allows estimates of 4-loop

terms. d/a is BLM scale - differs for each W, so see

running of .

Wlatt =3

V(d/a)

Quenched

Unquenched

!s HPQCD, Mason et al, hep-lat/0503005

I log W12 = −5.551αV (3/a)× [1− 0.86α+ 1.72α2−5(2)| z ↑

α3−1(2)| z ↑

α4 + . . .]

α = 0.25 . . . 0.4 fitted

IDetermining Parameters of QCD :

nf = 3

nf = 00.2

V(d/a)

2 4 6 8

d/a(G eV)

Combine 3-loop

pert. th. for 28

different

quantities (mainly

Wilson loops)

with numerical

calc. on lattice

Results at 3 values of a allows estimates of 4-loop

terms. d/a is BLM scale - differs for each W, so see

running of .

Wlatt =3

V(d/a)

Quenched

Unquenched

!s HPQCD, Mason et al, hep-lat/0503005

I log W12 = −5.551αV (3/a)× [1− 0.86α+ 1.72α2−5(2)| z ↑

α3−1(2)| z ↑

α4 + . . .]

α = 0.25 . . . 0.4 fitted

IDetermining parameters of QCD: mq

Use pert. th. to convert

lattice bare mass to

New b/c masses (GeV):

mb(mb) = 4.4(3)

mc(mc) = 1.10(13)

Light quark masses now

have 2-loop matching! 1

Quark masses (MeV)

2005 lattice QCD2004 PDG MS

HPQCD, Gray et al, hep-lat/0507013,

Nobes et al, LAT05;

N.B. 3-loop nf=2 result mb = 4.21(7) from

Direnzo and Scorzato, hep-lat/0408015

ms = 86(5); mu = 1.9(3); md = 4.5(4) in MeV at 2 GeV

HPQCD/MILC, Aubin et al, hep-lat/0405022; HPQCD, Mason, LAT05 updates this with 2-loop results

I what is the influence of the taste violations to these numbers?

I there is progress in NP renormalization −→ Knechtli

mb(mb) = 4.4(3)

mc(mc) = 1.10(13)

Quark masses (MeV)

Nobes et al, LAT05;

mb(mb) = 4.4(3)

mc(mc) = 1.10(13)

Quark masses (MeV)

Nobes et al, LAT05;

The future

is bright

I already now: large impact of LQCD, in particular onCKM physics −→ Lubiczdetermination of fundamental parameters −→ Knechtli

I continuous development of theoretical tools, conceptionally newideasLattice PT, NP Symanzik improvement, NP renormalization,physics from finite size effects, exact chiral symmetry, ChPTincluding lattice effects ...

I progress in algorithms small quark masses −→ Giusti– often driven more by intuition than by theoretical understanding

a very complex dynamical system −→ a challenge– work is not sufficiently appreciated

I driven also by advances in computer technology

I there is lots to do

The futureis bright

introduction to qcd on the lattice

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